A Finite Dimensional Control of the Dynamics of the...

Applied Mathematics & Information Sciences 3(2) (2009), 207–221– An International Journalc©2009 Dixie W Publishing Corporation, U. S. A.

A Finite Dimensional Control of the Dynamics of the Generalized

Korteweg-de Vries Burgers Equation

Nejib Smaoui1 and Mohamed Zribi2

1 Department of Mathematics and Computer Science, Kuwait University

P. O. Box 5969, Safat 13060, Kuwait

Email Address: [email protected] Department of Electrical Engineering, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait

Email Address: [email protected]

Received September 16, 2008; Revised January 17, 2009

The paper deals with the finite dimensional control of the generalized Korteweg-de Vries Burgers (GKdVB) partial differential equation (PDE). A Karhunen-LoeveGalerkin projection procedure is used to derive a system of ordinary differential equa-tions (ODEs) that mimics the dynamics of the GKdVB equation. Using Lyapunovtheory, it is shown that the highly nonlinear system of ODEs is stable. However, thesimulation results indicate that the system of ODEs converges slowly to the origin.Therefore, two control schemes are proposed for the system of ODEs; the main objec-tive of the controllers is to speed up the convergence to the origin. The first controlleris a linear state feedback controller whereas the second controller is a nonlinear con-troller. It is proven that both controllers guarantee the asymptotic convergence of thestates of the system of ODEs to zero. Simulation results indicate that the proposedcontrol schemes work well.

Keywords: Generalized Korteweg-de Vries Burgers equation, Karhunen-Loeve de-composition, linear control, nonlinear control.

1 Introduction

In this paper, we investigate the finite dimensional control of the dynamics of the gen-eralized Korteweg-de Vries-Burgers (GKdVB ) equation

∂u

∂t− ν

∂2u

∂x2+ µ

∂3u

∂x3+ uα ∂u

∂x= 0, x ∈ (0, 2π), t ≥ 0 (1.1)

208 Nejib Smaoui and Mohamed Zribi

with periodic boundary conditions

u (0, t) = u (2π, t) ,∂u

∂x(0, t) =

∂u

∂x(2π, t) ,

∂2u

∂x2(0, t) =

∂2u

∂x2(2π, t) , (1.2)

and the initial conditionu (0, t) = f (x) = sin(x), (1.3)

where µ, ν > 0 and α is a positive integer.When α = 1 in Eq. (1.1), the GKdVB equation reduces to the Korteweg-de Vries

Burgers (KdVB) equation which was used as a model for many physical phenomena andhydrodynamics processes. For instance, the KdVB was used as a model for long wavesin shallow water [9] and as a model of unidirectional propagation of planar waves [18].In the last few decades, the KdVB equation has been investigated analytically as well asnumerically, see references [1, 4, 6–8, 15, 18, 20, 25]. If α = 1 and ν = 0, the GKdVBequation reduces to the Korteweg-de Vries (KdV) equation derived by Korteweg and deVries as a model for waves propagating on the surface of a canal [19]. If α = 1 and µ = 0,the GKdVB equation reduces to the Burgers equation which models turbulent liquid flowthrough a channel [11].

Recently, a lot of work has been done to control the KdVB equation, the KdV equationand the Burgers equation. For example, Balogh and Krstic [4] investigated the controlproblem of the KdVB equation using boundary control. In their work, global stabilityof the solution in the L2-sense and global stability in the H1-sense were proved. Rosier[23, 24] worked on the the control problem of the KdV equation where an exact boundarycontrol of the linear and nonlinear KdV equations was established in [23]; and the controlwas illustrated numerically in [24]. Smaoui [29, 30] and Smaoui et al. [33] consideredthe boundary and distributed control of the Burgers equation. A boundary control is usedin [29] to show the exponential stability of the Burgers equation analytically as well asnumerically. In [30], a system of ODEs was constructed to mimic the dynamics of theBurgers equation, then a state feedback control scheme was implemented on the system toshow that the solution of the Burgers equation can be controlled to any desired state.

In this paper, the control problem of the GKdVB equation with periodic boundary con-ditions is investigated. A state feedback controller and a nonlinear controller are designedfor the GKdVB equation. Simulation results are presented to illustrate the developed the-ory.

In section 2, numerical simulations of the GKdVB equation are obtained using thepseudo-spectral Fourier Galerkin method. Then, the Karhunen-Loeve decomposition isused on the simulated data to extract the coherent structures of the equation for α = 2.Next, we present the Galerkin projection method and apply it on the GKdVB equation forα = 2; a system of ODEs that mimics the dynamics of the GKdVB equation is extracted.The data coefficients obtained are then used to approximate the numerical solution of the

A Finite Dimensional Control of the Dynamics of the Generalized KdVB Equation 209

GKdVB equation. The approximation is then compared to the numerical solution of theGKdVB equation. Section 3 gives the ODE dynamics which mimics the dynamics ofthe GKdVB and plots its numerical solution. Section 4 presents two control schemes tospeed up the convergence of the GKdVB equation; numerical simulations are presented toillustrate the presented theory. Finally, some concluding remarks are given in section 5.

2 The Karhunen-Loeve Galerkin Projection

2.1 The Karhunen-Loeve decomposition

The Karhunen-Loeve (K-L) decomposition is a very useful and powerful statisticaltechnique that is used in many applications; it has many different names depending on thefield where it is used in. It is known as the Hotelling transform in image processing [13,16],the principal component analysis (PCA) in pattern recognition and signal processing [17],the empirical component analysis in statistical weather prediction [21], the quasi-harmonicmodes in biology [10], the factor analysis in psychology and economics [14], and theproper orthogonal decomposition (POD) or the singular value decomposition (SVD) influid dynamics [22, 26].

Among the many applications that utilize the Karhunen-Loeve (K-L) decomposition,the K-L decomposition was used in the analysis of the two-dimensional Navier-Stokes (N-S) equation [2, 3, 28, 32], and in the study of flames [27, 31].

In this paper, we use the K-L decomposition on the numerically simulated data of theGKdVB equation given in (1) in order to extract the most energetic eigenfunctions or co-herent structures that span the data set in an optimal way. Since the K-L decompositionis a well known procedure, we refer the reader to the references cited above for a detaileddescription of the technique.

The solution u(x, t) of the GKdVB equation, obtained using a pseudo-spectral Galerkinmethod, can be written in terms of the eigenfunctions or coherent structures, ψ

′is, as

u (x, t) =M∑

i=1

ai (t)ψi (x) , (2.1)

where ai(t) are the data coefficients calculated from the projections of the sample vectorsolution onto an eigenfunction, i.e.,

ai =< u, ψi >

< ψi, ψi >, i = 1, . . . ,M

with M being the number of snapshots.Figure 2.1 shows the solution u (x, t) of the GKdVB equation as it evolves to its steady

state solution. Applying the K-L decomposition on the numerical solution presented in


02

46

810

0

2

4

6

8−1

−0.5

0

0.5

1

tx

u(x,

t)

Figure 2.1: A 3-D landscape plot of the numerical solution of the GKdVB equation when α = 2,ν = 0.5, and µ = 0.01.

0 2 4 6 8−4

−2

0

2

4

ψ1 versus x

x

ψ1

0 2 4 6 8−0.4

−0.2

0

0.2

0.4

ψ2 versus x

x

ψ2

−4 −2 0 2 4−0.4

−0.2

0

0.2

0.4

ψ2 versus ψ

1

ψ1

ψ2

Figure 2.2: The eigenfunctions associated with the solution of the GKdVB equation when α = 2,ν = 0.5, and µ = 0.01.


0 5 100

0.1

0.2

0.3

0.4a1 vs time

t, sec

a1(t

)

0 5 10−0.6

−0.4

−0.2

0

0.2a2 vs time

t, sec

a2(t

)

0 0.1 0.2 0.3 0.4−0.6

−0.4

−0.2

0

0.2a2 vs a1

a1(t)

a2(t

)

Figure 2.3: The data coefficients associated with the eigenfunctions of the solution of the GKdVBequation when α = 2, ν = 0.5, and µ = 0.01.

02

46

810

0

2

4

6

8−1

−0.5

0

0.5

1

tx

u(x,

t)

Figure 2.4: A 3-D landscape plot of the approximated solution of the GKdVB equation when α = 2,ν = 0.5, and µ = 0.01 .


Figure 2.1, two eigenfunctions capturing 99.98% of the energy were obtained; these eigen-functions are plotted in Figure 2.2. The first eigenfunction captures 99.54% of the energyand the second one captures 0.44% of the energy. Figure 2.3 presents the correspondingdata coefficients, and Figure 2.4 depicts the approximated solution of the GKdVB equationusing the above eigenfunctions.

When comparing Figures 2.1 and 2.4, one can conclude that the K-L decompositionwas able to capture the large scale dynamics of the GKdVB equation with only two eigen-functions.

2.2 The Galerkin projection

The Galerkin projection is a method used to replace infinite dimensional continuoussystems by finite dimensional ones [5, 12]. In this section, the Galerkin projection is usedto extract a system of ODEs from the GKdVB equation. The system of ODEs can be solvedto get an approximate solution of the original PDE. Thus, the approximation can be writtenin the following form:

u (x, t) ≈K∑

i=1

ai (t)ψi (x) , (2.2)

where ai (t) is the ith solution of the system of ODEs and can be computed in a waythat minimize the residual error produced by the approximate solution above and ψi (x)is the ith eigenfunction from the K-L decomposition; K was taken to be 2, since twoeigenfunctions capture most of the dynamics of the GKdVB equation.

In order to extract the system of ODEs, we first write the original PDE as

∂u

∂t= D (u) , (2.3)

with given initial and boundary conditions, where “D” is the differential operator. Then,the system of ODEs is derived by projecting the normalized eigenfunctions, onto the PDEas

ai (t) =⟨

D( K∑

i=1

ai (t)ψi (x)), ψi (x)

⟩, i = 1, . . . , K (2.4)

with initial condition

ai (0) = 〈u (x, 0) , ψi (x)〉, i = 1, . . . ,K, (2.5)

where u (x, 0) is obtained from the original PDE.Using Eq.(2.2) and taking into account the two most energetic eigenfunctions, the

GKdVB equation (1.1) with α = 2

∂u

∂t= ν

∂2u

∂x2− µ

∂3u

∂x3− u2 ∂u

∂x,


becomes2∑

i=1

ai (t)ψi (x) = ν

2∑

i=1

ai (t) ψi′′ (x)− µ

2∑

i=1

ai (t) ψi′′′ (x)

−( 2∑

i=1

ai (t)ψi (x))2( 2∑

i=1

ai (t)ψ′i (x)), (2.6)

or2∑

i=1

ai (t)ψi (x) = ν2∑

i=1

ai (t)ψi′′ (x)− µ

2∑

i=1

ai (t) ψi′′′ (x)

−2∑

i=1

2∑

j=1

a2i (t) aj (t)ψ2

i (x)ψ′j (x)

− 22∑

i 6=j=1

ai (t) a2j (t)ψi (x) ψj (x)ψ′j (x) , (2.7)

where ai (t) is the derivative with respect to time and ψ′i (x) is the derivative with respect

to x. Now, taking the Euclidean inner product of the above equation with ψk, k = 1, 2 andusing the orthogonality property of ψ’s, we obtain the following system of ODEs

ak (t) = ν

2∑

i=1

ai (t) 〈ψk, ψi′′〉 − µ

2∑

i=1

ai (t) 〈ψk, ψi′′′〉

−2∑

i=1

2∑

j=1

a2i (t) aj (t) 〈ψk, ψ2

i ψ′j〉

− 22∑

i6=j=1

ai (t) a2j (t) 〈ψk, ψiψjψ

′j〉, (k = 1, 2). (2.8)

The solution to the above ODE system can be obtained numerically using any ODE solver.

3 Stability of the ODE Dynamics of the GKdVB Equation

We have applied the above procedure on the GKdVB equation when α = 2 and withthe initial condition given by Eq.(1.3). The model of the obtained system of ODEs can bewritten as

a1 = f1(a1, a2),

a2 = f2(a1, a2), (3.1)

where

f1(a1, a2) = −p1a21a2 − p2a1a

22 − p3a

32 + p4µa2 − p5νa1 + p6νa2, (3.2)


f2(a1, a2) = p2a21a2 + p3a1a

22 + p1a

31 − p4µa1 + p6νa1 − p7νa2 (3.3)

with

p1 = 0.4899793644, p2 = 0.0570998562, p3 = 0.5148487537,

p4 = 1.0208303477, p5 = 1.0043594987, p6 = 0.0043221400,

p7 = 1.0140806414, for α = 2, µ = 0.01 and ν = 0.5.

Proposition 3.1. The dynamics of the system of ODEs of the GKdVB equation given by(3.1) is asymptotically stable.

Proof. Let the Lyapunov function candidate V be such that

V (a1, a2) =12a21 +

12a22. (3.4)

Taking the derivative of V with respect to time, it follows that

V = a1a1 + a2a2

= (−p1a21a2 − p2a1a

22 − p3a

32 + p4µa2 − p5νa1 + p6νa2)a1

+ (p2a21a2 + p3a1a

22 + p1a

31 − p4µa1 + p6νa1 − p7νa2)a2

= −p5νa21 − p7νa2

2 + 2p6νa1a2

= −p5νa21 − p7ν

[(a2 − p6

p7a1)2 − p2

6

p27

a21

]

= −(p5 − p26

p7)νa2

1 − p7ν(a2 − p6

p7a1)2

< 0 for (a1, a2) 6= (0, 0). (3.5)

Note that V is negative definite because it can be easily checked that p5 − p26/p7 > 0, and

p7 and ν are positive. Therefore, the dynamics of the system of ODEs given by (3.1) isasymptotically stable.

The system of ODEs given by (3.1) is simulated using the Matlab software. The initialconditions are chosen to be a1(0) = 10 and a2(0) = −10.

Figure 3.1 shows the simulation results of the system of ODEs. The coefficients a1(t)versus a2(t) when the time t is taken to be 10 seconds are plotted. Note that a1(t) and a2(t)take about 5 seconds to converge to zero. Therefore, it can be concluded that the dynamicsof the ODEs of the GKdVB equation is asymptotically stable.

4 Design of Controllers for the GKdVB Equation

To speed up the convergence of the coefficients a1(t) and a2(t) to zero, we propose toadd a forcing term to the system of equations. The forced system of ODEs can be written


0 5 10−20

−10

0

10

20a1 vs time

t, sec

a1(t

)

0 5 10−20

−10

0

10

20a2 vs time

t, sec

a2(t

)

−20 −10 0 10 20−20

−10

0

10

20a2 vs a1

a1(t)

a2(t

)

Figure 3.1: The profiles of the data coefficients a1(t) and a2(t) with no control.

as

a1 = −p1a21a2 − p2a1a

22 − p3a

32 + p4µa2 − p5νa1 + p6νa2 + b1u = f1(a1, a2) + b1u,

a2 = p2a21a2 + p3a1a

22 + p1a

31 − p4µa1 + p6νa1 − p7νa2 + b2u = f2(a1, a2) + b2u,

(4.1)

where u(t) is the forcing term (the input) of the system and b1 and b2 are multiplyingfactors.

It is desired to design control schemes such that the coefficients a1(t) and a2(t) con-verge to (0,0) as fast as possible.

4.1 Design of a state feedback controller

Let the design parameters k1 and k2 be chosen such that k1b2 = k2b1, k1b1 > 0 andk2b2 > 0.

Proposition 4.1. The state feedback controller

u(t) = −k1a1(t)− k2a2(t) (4.2)

guarantees the asymptotic stability of the forced system of ODEs given in (4.1).


V (a1, a2) =12a21 +

12a22. (4.3)


Taking the derivative of V with respect to time, it follows that,

V = a1a1 + a2a2

= (−p1a21a2 − p2a1a

22 − p3a

32 + p4µa2 − p5νa1 + p6νa2 + b1u)a1

+(p2a21a2 + p3a1a

22 + p1a

31 − p4µa1 + p6νa1 − p7νa2 + b2u)a2

= −p5νa21 − p7νa2

2 + 2p6νa1a2 + (b1a1 + b2a2)u

= −(p5 − p26

p7)νa2

1 − p7ν(a2 − p6

p7a1)2 + (b1a1 + b2a2)u

≤ (b1a1 + b2a2)(−k1a1 − k2a2)

= −b1k1a21 − b2k2a

22 − (b2k1 + b1k2)a1a2

= −b1k1a21 − b2k2

[a22 +

b2k1 + b1k2

b2k2a1a2

]

= −b1k1a21 − b2k2

[a2 +

b2k1 + b1k2

2b2k2a1

]2

+(b2k1 + b1k2)2

4b2k2a21

= −b2k2

[a2 +

b2k1 + b1k2

2b2k2a1

]2

+(b2k1 − b1k2)2

4b2k2a21

= −b2k2

[a2 +

b2k1 + b1k2

2b2k2a1

]2

.

Clearly V < 0 for (a1, a2) 6= (0, 0) and V = 0 for (a1, a2) = (0, 0). Therefore, V isnegative definite, and V is a Lyapunov function for the system (4.1). Hence, the system ofODEs given by (4.1) is asymptotically stable.

The forced system of ODEs given by (4.1) with the controller given by (4.2) is simu-lated using the Matlab ODE solver. The parameters b1 and b2 are taken to be b1 = b2 = 1.The initial conditions are chosen to be a1(0) = 10 and a2(0) = −10.

Figure 4.1 shows the simulation results of the forced system of ODEs when the statefeedback controller is used. In Figure 4.1, the plots of the coefficients a1(t) and a2(t)versus time and the plots of the coefficients a1(t) versus a2(t) are shown for three differentcases. The first case corresponds to gains k1 = k2 = 3; the second case, corresponds togains k1 = k2 = 10; and the third case corresponds to gains k1 = k2 = 20. It is clearfrom the figure that the larger the gains of the controller, the faster the coefficients a1(t)and a2(t) converge to zero. However, the increase in the convergence speed comes at theexpense of larger control actions.

4.2 Design of a nonlinear controller

Proposition 4.2. Let W be a positive scalar. The nonlinear controller

u(t) = −Wsign(b1a1(t) + b2a2(t)) (4.4)

guarantees the asymptotic stability of the forced system of ODEs given by (4.1).


0 1 2 3−15

−10

−5

0

5

10

15a1 vs time

t, sec

a1(t

)

0 1 2 3−15

−10

−5

0

5

10

15a2 vs time

t, sec

a2(t

)

−20 −10 0 10 20−20

−10

0

10

20a1 vs a2

a1(t)

a2(t

)

Figure 4.1: The profiles of the data coefficients a1(t) and a2(t) with the state feedback controller;the solid line corresponds to gains k1 = k2 = 3; the ‘- - -’ line corresponds to gains k1 = k2 = 10;and the bold line corresponds to gains k1 = k2 = 20.


V (a1, a2) =12a21 +

12a22. (4.5)

Taking the derivative of V with respect to time, it follows that

V = a1a1 + a2a2

= (−p1a21a2 − p2a1a

22 − p3a

32 + p4µa2 − p5νa1 + p6νa2 + b1u)a1

+(p2a21a2 + p3a1a

22 + p1a

31 − p4µa1 + p6νa1 − p7νa2 + b2u)a2

= −p5νa21 − p7νa2

2 + 2p6νa1a2 + (b1a1 + b2a2)u

= −(p5 − p26

p7)νa2

1 − p7ν(a2 − p6

p7a1)2 + (b1a1 + b2a2)u

≤ −W (b1a1 + b2a2)sign(b1a1 + b2a2).

Clearly V < 0 for (a1, a2) 6= (0, 0) and V (0, 0) = 0. Therefore, V is negative definiteand V is a Lyapunov function for the system (4.1) with controller 4.4. Thus, the controllergiven by (4.4) guarantees the asymptotic stability of the forced system given by (4.1).

Remark 4.1. It can be shown that using the hyperbolic tangent instead of the sign functionin (4.4) will also give good results (i.e., u = −Wtanh(b1a1 + b2a2)).


0 2 4 6−15

−10

−5

0

5

10

15a1 vs time

t, sec

a1(t

)

0 2 4 6−15

−10

−5

0

5

10

15a2 vs time

t, sec

a2(t

)

−10 0 10 20−10

−5

0

5

10

15a2 vs a1

a1(t)

a2(t

)

Figure 4.2: The profiles of the data coefficients a1(t) and a2(t) with the nonlinear controller, thesolid line corresponds to W = 100; the ‘- - -’ line corresponds to W = 500; and the bold linecorresponds to W = 1000.

The forced system of ODEs given by (3.6) with the controller given by (4.4) is simu-lated using the Matlab ODE solver. The parameters b1 and b2 are taken to be b1 = b2 = 1.The initial conditions are chosen to be a1(0) = 10 and a2(0) = −10. The hyperbolictangent is used in the simulations.

Figure 4.2 shows the simulation results of the system of ODEs when the proposednonlinear controller is used. In Figure 4.2, the coefficients a1(t) and a2(t) versus timeand the coefficients a1(t) versus a2(t) are shown for three cases corresponding to threedifferent gains, W = 100, W = 500, and W = 1000. Hence, it is clear from the figurethat these data coefficients are controlled to perform in a desired manner. Also, one can seefrom the figure that the larger the gain W of the controller, the faster the coefficients a1(t)and a2(t) converge to zero. Obviously, the larger the gain W the bigger the control actionsare going to be.

Remark 4.2. The linear and nonlinear controllers are designed to drive the coefficientsa1(t) and a2(t) to (0, 0). By using proper change of variables, it can be easily shown thatthe steady states values of a1 and a2 can take any desired values.


5 Concluding Remarks

This paper addresses the control problem of the GKdVB equation. The numerical so-lutions of the nonlinear PDE is obtained using a pseudo-spectral Galerkin method. Then,coherent structures of the solutions are obtained using the K-L decomposition method. Itis shown that for the case when α = 2, only two eigenfunctions are necessary to capturethe large scale dynamics of the equation. Applying the K-L Galerkin projection of the nu-merical solution data on the most energetic eigenfunctions results in a system of ODEs thatmimics the dynamics of the original PDE. The obtained system of ODEs is stable but itconverges slowly to (0, 0). A linear and a nonlinear control schemes are proposed to speedup the convergence to the steady states. It is proven that both control schemes guaranteethe asymptotic stability of the states of the forced system of ODEs. Moreover, numericalsimulations show the effectiveness of proposed controllers.

It should be mentioned that, even though the paper concentrates on the case of α = 2,similar results can be obtained for other integer values of α.

For future research, we will address the design of other controller such as adaptive andoptimal control schemes to the GKdVB equation and for different integer values of α.

References

[1] A. Armaou and P. D. Christofides, Wave suppression by nonlinear finite-dimensionalcontrol, Proceedings of the American Control Conference, San Diego, California,1999, 1091–1095.

[2] D. Armbruster, R. Heiland, E. Kostelich, and B. Nicolaenko, Phase-space analysis ofbursting behavior in Kalmogorov flow, Physica D 58 (1992), 392–401.

[3] D. Armbruster, B. Nicolaenko, N. Smaoui, and P. Chossat, Symmetries and dynamicsfor 2-D Navier-Stokes flow, Physica D 95 (1996), 81–93.

[4] A. Balogh and M. Krstic, Boundary control of the Korteweg-de Vries-Burgers equa-tion: Further results on stabilization and well-posedness, with numerical demonstra-tion, IEEE Transactions on Automatic Control 45 (2000), 1739–1745.

[5] G. Berkooz, and E. S. Titi, Galerkin projections and the proper orthogonal decompo-sition for equivariant equations, Phys. Lett. A 174 (1993), 94–102.

[6] P. Biler, Asymptotic behavior in time of solutions to some equations generalizingthe Korteweg-de Vries-Burgers equation, Bull. Polish Acad. Sciences, Mathematics,3 (1984), 275–282.

[7] J. L. Bona and L. Luo, Decay of solutions to nonlinear dispersive wave equations,Differential Int. Equations, 6 (1993), 961–980.

[8] J.L Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation,Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555–601.


[9] V. Boyko, On new generalizations of the Burgers and Korteweg-de Vries equations,Symmetry in Nonlinear Mathematics Physics 1 (1997), 122–129.

[10] C. L. Brooks, M. Karplus, and B. M. Pettitt, A Theoretical Prespective of Dynamics,Structure and Thermodynamics, New York, NY: Wiley, 1988.

[11] T. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl.Mech. 1 (1948), 171–199.

[12] D. K. Cheng and C.-H. Liang, Thinning technique for moment-method solutions,Proc. of the IEEE 71 (1983), 265–266.

[13] A. C. Gonzalez and P. Winiz, Digital Image Processing, 2nd Edition, Addison-Wesley, Reading, MA, 1987.

[14] H. Harman, Modern Factor Analysis, Chicago: University of Chicago, 1960.[15] N. Hayashi, E. I. Kaikina, and I. A. Shishmarev, Asymptotics of solutions to the

boundary-value problem for the Korteweg-de Vries-Burgers equation in a half-line, J.Math. Anal. Appl. 265 (2002), 343–370.

[16] H. Hotelling, Analysis of complex of statistical variables into principal components,Journal of Educational Psychology, 24 (1933), 498–520.

[17] L.T. Jollife, Principal Component Analysis New York, NY: Springer Verleg, 1986.[18] G. Karch, Self-similar large time behavior of solutions to Korteweg-de Vries-Burgers

equation, Nonlinear Analysis 35 (1999), 199–219.[19] D. J. Korteweg and G. de Vries, On the change form of long waves advancing in a

rectangular canal and on a new type of long stationary wave, Phil. Mag. 39 (1895),422–443.

[20] W.-J. Liu, and M. Krstic, Controlling nonlinear water waves: Boundary stabilizationof the Korteweg-de Vries-Burgers equation, Proceedings of the American ControlConference, San Diego, California, 1999, 1637–1641.

[21] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963), 130–141.[22] J. L. Lumley, Atmospheric Turbulence and Radio Wave propagation. in: A. M. Ya-

glom, V. I. Tatarski (Eds.), Nauka, Moscow, 1967.[23] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a

bounded domain, ESAIM Conrol optim. Calc. Var. 2 (1997), 33–55 (electronic).[24] L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation -

A numerical study, ESAIM Proceedings 4 (1988), 255–267.[25] D.L. Russel and B.Y. Zhang, Exact controllability and stability of the Korteweg-de

Vries equation, Trans. Amer. Math. Soc. 348 (1996), 3643–3672.[26] L. Sirovich, Turbulence and dynamics of coherent structures. Part I. coherent struc-

tures, Quart. Appl. Math. 45 (1987), 561–571.[27] N. Smaoui, Artificial neural network-based low-dimensional model for spatio-

temporally varying cellular flames, Appl. Math. Modeling 21 (1997), 739–748.[28] N. Smaoui, N., A model for the unstable manifold of the bursting behavior in the 2-d

Navier-Stokes flow, SIAM J. Sci. Comput. 23 (2001), 824–840.[29] N. Smaoui, Nonlinear boundary control of the generalized Burgers equation, Nonlin-

ear Dynamics Journal 37 (2004), 75–86.


[30] N. Smaoui, Boundary and distributed control of the viscous Burgers equation, Journalof Computational and Applied Mathematics 182 (2005), 91–104.

[31] N. Smaoui and S. Al-Yakoob, Analyzing the dynamics of cellular flames usingKarhunen-Loeve decomposition and associative neural networks, SIAM J. Sci. Com-put. 24 (2003), 1790–1808.

[32] N. Smaoui and D. Armbruster, Symmetry and the Karhunen-Loeve analysis, SIAM J.Sci. Comput. 18 (1997), 1526–1532.

[33] N. Smaoui, M. Zribi, and A. Almulla, Sliding Mode Control of the Forced Gener-alized Burgers Equation, IMA Journal of Mathematical Control & Information 23(2006), 301–323.

Nejib Smaoui obtained the B.Sc. degree in Mechanical Engineeringfrom Iowa State University, Ames, Iowa in 1987. He received the M.Sc.degree in Mechanical Engineering, the M.Sc. degree in Mathematics andthe Ph.D. degree in Mathematics from Arizona State University, Tempe,Arizona in 1989, 1990, and 1994, respectively.

He is currently a Professor in Mathematics in the Department of Math-ematics and Computer Science and Assistant Vice President for Research at Kuwait Uni-versity. His research interests include dynamical systems, chaos, differential equations andcontrol.

Mohamed Zribi obtained the B.S. degree in Electrical Engineering(Magna Cum Laude Honors) in 1985 from the University of Houston,Houston, Texas. He received the M.S.E.E Degree and the Ph.D. Degreein Electrical Engineering in 1987 and 1992 from Purdue University, WestLafayette, Indiana.

He held a faculty position in the School of Electrical Engineering,Nanyang Technological University, Singapore from 1992 till 1998. He is currently a Pro-fessor in the Electrical Engineering Department, College of Engineering & Petroleum,Kuwait University. His research interests are in nonlinear control, adaptive control, controlapplications and robotics.

A Finite Dimensional Control of the Dynamics of the...

Documents

Transcript of A Finite Dimensional Control of the Dynamics of the...